Theorem imaeq2 4925

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4862	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4816	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4600	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4600	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2215	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1335   ran crn 4588   |` cres 4589   "cima 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-cnv 4595  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600

This theorem is referenced by:  imaeq2i  4927  imaeq2d  4929  ssimaex  5530  ssimaexg  5531  isoselem  5771  f1opw2  6027  fopwdom  6782  ssenen  6797  fiintim  6874  fidcenumlemrk  6899  fidcenumlemr  6900  sbthlem2  6903  isbth  6912  ennnfonelemp1  12177  ennnfonelemnn0  12193  ctinfomlemom  12198  ctinfom  12199  tgcn  12650  iscnp4  12660  cnpnei  12661  cnima  12662  cnconst2  12675  cnrest2  12678  cnptoprest  12681  txcnp  12713  txcnmpt  12715  metcnp3  12953